An Analogy of the Carleson-Hunt Theorem with Respect to Vilenkin Systems

In this paper we discuss and prove an analogy of the Carleson-Hunt theorem with respect to Vilenkin systems. In particular, we use the theory of martingales and give a new and shorter proof of the almost everywhere convergence of Vilenkin-Fourier series of f is an element of L-p(G(m)) for p > 1 in case the Vilenkin system is bounded. Moreover, we also prove sharpness by stating an analogy of the Kolmogorov theorem for p = 1 and construct a function f is an element of L-1(G(m)) such that the partial sums with respect to Vilenkin systems diverge everywhere.

Automated summary

In this paper, we discuss and prove an analogy of the Carleson-Hunt theorem with respect to Vilenkin systems. We use the theory of martingales to provide a new and shorter proof of the almost everywhere convergence of Vilenkin-Fourier series for p > 1, in case the Vilenkin system is bounded. Moreover, we also prove the sharpness by stating an analogy of the Kolmogorov theorem for p = 1 and construct a function in L-1(G(m)) such that the partial sums with respect to Vilenkin systems diverge everywhere.